Module 2: Metamorphosis Biochemistry

Metamorphosis is one of the most dramatic transformations in biology: a crawling caterpillar dissolves its own tissues and reconstructs itself as a flying butterfly. This module examines the hormonal control systems, the physics of morphogen gradients in imaginal discs, and the biochemistry of tissue remodeling — all from a quantitative, biophysical perspective.

1. Complete vs Incomplete Metamorphosis

Insect metamorphosis comes in two major forms, each representing a different evolutionary strategy for development and resource partitioning.

Holometabolous (Complete)

Stages: Egg -> Larva -> Pupa -> Adult

Orders: Coleoptera (beetles), Diptera (flies), Lepidoptera (moths/butterflies), Hymenoptera (ants/bees/wasps)

The pupal stage is a radical innovation: the larva is essentially a feeding machine with a fundamentally different body plan from the adult. During pupation, most larval tissues are histolyzed (broken down) and rebuilt from imaginal disc precursor cells.

~85% of all insect species are holometabolous.

Hemimetabolous (Incomplete)

Stages: Egg -> Nymph -> Adult (no pupa)

Orders: Orthoptera (grasshoppers), Hemiptera (true bugs), Odonata (dragonflies), Blattodea (cockroaches)

Nymphs resemble miniature adults. Development is gradual through successive molts (instars). Wing pads grow progressively. No pupal remodeling phase.

Ancestral condition; holometaboly evolved from this ~350 Ma in the Carboniferous.

Evolutionary Advantage of Complete Metamorphosis

The key advantage is niche partitioning: larvae and adults exploit entirely different ecological niches, eliminating intraspecific competition between life stages. A caterpillar eats leaves; the butterfly drinks nectar. A maggot decomposes organic matter; the adult fly feeds on sugar solutions. This doubles the ecological opportunities available to a single species and is thought to explain why holometabolous orders contain far more species than hemimetabolous ones.

2. Hormonal Control of Metamorphosis

Insect metamorphosis is controlled by two master hormones whose ratio determines the developmental outcome of each molt. The system is elegant in its simplicity: a binary switch controlled by two hormones.

Juvenile Hormone (JH)

Source: Corpora allata (paired endocrine glands behind the brain)

Chemistry: Sesquiterpenoid (C15 or C16). JH III is the most common form. Lipophilic, transported by hemolymph binding proteins.

Function: “Status quo” hormone. Maintains larval identity. Prevents metamorphic gene expression. High JH = “stay as larva.”

Receptor: Methoprene-tolerant (Met) protein, a bHLH-PAS transcription factor. JH-Met complex activates Kr-h1 (anti-metamorphic gene).

20-Hydroxyecdysone (20-HE)

Source: Prothoracic glands (stimulated by PTTH from the brain)

Chemistry: Steroidal hormone (polyhydroxylated ketosteroid). Ecdysone is hydroxylated to active 20-HE in peripheral tissues.

Function: Molting hormone. Triggers apolysis (separation of old cuticle), synthesis of new cuticle, and ecdysis (shedding). Also triggers metamorphic gene cascade when JH is absent.

Receptor: EcR-USP heterodimer (nuclear receptor). Activates hierarchical transcription factor cascade: EcR -> E74 -> E75 -> downstream.

2.1 The JH/Ecdysone Decision Matrix

The developmental fate at each molt is determined by the combinatorial logic of JH and ecdysone (20-HE) levels:

High JH + 20-HE Pulse

Larval-Larval Molt

Larva grows, sheds cuticle, remains larva

Low JH + 20-HE Pulse

Larval-Pupal Molt

Triggers pupation; metamorphic genes activated

No JH + 20-HE Pulse

Pupal-Adult Eclosion

Adult differentiation; final emergence

Mathematical model of JH/ecdysone interaction:

\(\text{Developmental fate} = f\!\left(\frac{[\text{JH}]}{K_{\text{JH}}}\right) \cdot g\!\left(\frac{[\text{20-HE}]}{K_{\text{E}}}\right)\)

where \(f\) is a Hill function for JH-dependent gene suppression:

\(f(x) = \frac{1}{1 + x^n}, \quad g(x) = \frac{x^m}{1 + x^m}\)

with Hill coefficients \(n \approx 3{-}4\) (sharp JH threshold) and\(m \approx 2\) (cooperative ecdysone binding). The high Hill coefficient for JH creates a switch-like response: small changes in JH titer trigger abrupt developmental transitions.

2.2 PTTH and the Critical Weight Checkpoint

Prothoracicotropic hormone (PTTH) from brain neurosecretory cells triggers ecdysone release. PTTH secretion is gated by a critical weight checkpoint: the larva must reach a minimum size before the brain permits the final ecdysone pulse that triggers pupation. This ensures the adult will be viable.

Critical weight threshold (empirical):

\(W_{\text{crit}} \approx 0.5 \cdot W_{\text{final}} \quad \text{(in Manduca sexta)}\)

Below critical weight: JH clearance is blocked, preventing pupation. Above critical weight: JH degradation proceeds, corpora allata activity decreases, and the next ecdysone pulse triggers metamorphosis.

3. Imaginal Discs and Morphogen Gradients

In holometabolous insects, adult structures develop from imaginal discs — small clusters of undifferentiated, diploid cells set aside during embryogenesis. While the larva grows and feeds, these discs remain quiescent (in Drosophila, each disc starts as ~10-50 cells and grows to ~50,000 cells by the late third instar). During pupation, the discs evert and differentiate into adult organs: wings, legs, eyes, antennae, and genitalia.

Imaginal Disc Inventory (Drosophila)

Eye-Antennal

1 pair

Wing

1 pair

Haltere

1 pair

Leg

3 pairs

Labial

1 pair

Clypeolabral

Genital

Total

19 discs

3.1 Morphogen Gradient Patterning

Each imaginal disc is patterned by morphogen gradients that specify cell fates based on distance from a source. The key morphogens are Decapentaplegic (Dpp, a BMP homolog) and Wingless (Wg, a Wnt homolog), which are secreted from the anterior-posterior and dorsal-ventral boundaries respectively.

Morphogen concentration profile (steady-state diffusion + degradation):

\(\frac{\partial C}{\partial t} = D\nabla^2 C - kC + S(x)\)

where \(D\) is the effective diffusion coefficient, \(k\) is the first-order degradation rate, and \(S(x)\) is the source term (localized at the compartment boundary).

For a localized point source at \(x = 0\) in one dimension, the steady-state solution is an exponential gradient:

Morphogen Gradient (Exponential Profile):

\(C(x) = C_0 \cdot \exp\!\left(-\frac{|x|}{\lambda}\right), \quad \lambda = \sqrt{\frac{D}{k}}\)

where \(C_0\) is the concentration at the source, and\(\lambda\) is the decay length. For Dpp in the Drosophila wing disc:\(D \approx 0.1\;\mu\text{m}^2/\text{s}\),\(k \approx 10^{-4}\;\text{s}^{-1}\), giving \(\lambda \approx 30\;\mu\text{m}\) (~8 cell diameters).

French Flag model: threshold-based fate specification

Cells read the local morphogen concentration and adopt fates based on thresholds:

\(\text{Fate} = \begin{cases} \text{Vein (high)} & C > \theta_1 \\ \text{Intervein (medium)} & \theta_2 < C < \theta_1 \\ \text{Margin (low)} & C < \theta_2 \end{cases}\)

The precision of patterning depends on \(|\nabla C| / \sigma_C\) at the threshold — the gradient steepness relative to noise. The exponential profile ensures constant relative precision: \(|\nabla C|/C = 1/\lambda\) everywhere.

4. Histolysis and Histogenesis

During pupation, the larval body undergoes two concurrent processes: histolysis (tissue breakdown) and histogenesis (adult tissue construction). This is not mere remodeling — it is a near-total deconstruction and reconstruction.

Histolysis (Breakdown)

Autophagy: Larval cells activate self-digestion pathways. Autophagosomes engulf organelles and cytoplasm, delivering them to lysosomes for recycling. Triggered by ecdysone via the Atg1/ULK1 kinase complex.

Apoptosis: Programmed cell death via caspase cascade. Ecdysone activates the reaper/hid/grim pro-apoptotic genes, which inhibit DIAP1 (inhibitor of apoptosis), releasing the initiator caspase Dronc and effector caspase DrICE.

Phagocytosis: Hemocytes (immune cells) clear cellular debris. Macrophage-like plasmatocytes recognize phosphatidylserine on apoptotic cells.

Histogenesis (Construction)

Imaginal disc eversion: Discs unfold from their invaginated position and expand to form adult appendages. Cell shape changes (apical constriction) drive the physical eversion process.

Cell differentiation: Morphogen gradients (established earlier) now drive terminal differentiation. Wing disc cells differentiate into vein, intervein, sensory bristle, or margin cell types.

Cuticle secretion: Newly differentiated epidermal cells secrete the adult cuticle, which undergoes sclerotization and pigmentation after eclosion.

4.1 Caspase Activation Cascade Kinetics

The apoptotic caspase cascade during histolysis follows a switch-like activation pattern. The initiator caspase (Dronc) cleaves and activates effector caspases (DrICE, Dcp-1), which amplify the signal through positive feedback:

Caspase activation dynamics (simplified two-component model):

\(\frac{d[\text{DrICE}^*]}{dt} = k_1[\text{Dronc}^*]\cdot[\text{DrICE}] + k_2\frac{[\text{DrICE}^*]^2}{K_m^2 + [\text{DrICE}^*]^2}\cdot[\text{DrICE}] - k_d[\text{DrICE}^*]\)

The first term is initiator-mediated activation. The second term represents positive feedback (Hill coefficient = 2) where active DrICE promotes its own activation (through cleavage of DIAP1). The third term is degradation/inhibition. This creates a bistable switch: once activated, the cascade is irreversible.

Bistability condition:

\(\frac{k_2}{k_d} > \frac{4K_m}{[\text{DrICE}]_{\text{total}}}\)

When this condition is met, the system has two stable states (alive/dead) separated by an unstable threshold. This ensures apoptosis is an all-or-nothing decision — cells do not partially die.

Hormonal Control Through Development

JH and 20-HE levels through the complete metamorphic life cycle of a holometabolous insect, showing how their ratio determines developmental transitions.

Interactive Simulations

Simulation 1: JH and Ecdysone Levels Through Development

Models the temporal dynamics of juvenile hormone and 20-hydroxyecdysone through five larval instars, pupation, and adult eclosion.

Hormonal Dynamics of Insect Metamorphosis

Python

script.py161 lines

import numpy as np
import matplotlib.pyplot as plt
import matplotlib
matplotlib.use('Agg')

# Time in arbitrary developmental units (0 = egg hatch, 100 = adult eclosion)
t = np.linspace(0, 100, 2000)

def ecdysone_pulse(t, t_center, width=1.5, amplitude=1.0):
    """Generate a single ecdysone pulse (Gaussian)"""
    return amplitude * np.exp(-(t - t_center)**2 / (2 * width**2))

# Build ecdysone profile: pulse at each molt
# L1->L2 at t=12, L2->L3 at t=25, L3->L4 at t=38, L4->L5 at t=50
# Pupal commitment at t=58, Pupal molt at t=65, Eclosion at t=85
ecdysone = (ecdysone_pulse(t, 12, 2.0, 0.7) +
            ecdysone_pulse(t, 25, 2.0, 0.8) +
            ecdysone_pulse(t, 38, 2.0, 0.85) +
            ecdysone_pulse(t, 50, 2.5, 0.9) +
            ecdysone_pulse(t, 58, 2.0, 1.0) +  # pupal commitment
            ecdysone_pulse(t, 72, 3.0, 0.6) +   # mid-pupal
            ecdysone_pulse(t, 85, 2.5, 1.0))     # eclosion

# Build JH profile: high in larval stages, drops before pupation
jh_base = 0.8 / (1 + np.exp(0.2 * (t - 52)))  # sigmoidal decline
# Add small fluctuations correlated with ecdysone pulses
jh = jh_base + 0.15 * np.sin(2 * np.pi * t / 15) * (t < 50).astype(float) * 0.5
jh = np.clip(jh, 0, 1)

fig, axes = plt.subplots(3, 1, figsize=(14, 12))

# ---- Panel 1: Hormone levels ----
ax = axes[0]
ax.fill_between(t, 0, jh, alpha=0.2, color='#4ade80')
ax.plot(t, jh, color='#4ade80', linewidth=2.5, label='Juvenile Hormone (JH)')
ax.fill_between(t, 0, ecdysone, alpha=0.2, color='#fbbf24')
ax.plot(t, ecdysone, color='#fbbf24', linewidth=2.5, label='20-Hydroxyecdysone')

# Stage boundaries
stage_bounds = [0, 12, 25, 38, 50, 58, 65, 85, 100]
stage_names = ['L1', 'L2', 'L3', 'L4', 'L5', 'Pre-\npupa', 'Pupa', 'Adult']
stage_colors = ['#4ade80', '#4ade80', '#4ade80', '#4ade80', '#4ade80', '#fbbf24', '#f97316', '#22d3ee']

for i, (start, end) in enumerate(zip(stage_bounds[:-1], stage_bounds[1:])):
    mid = (start + end) / 2
    ax.text(mid, 1.05, stage_names[i], ha='center', fontsize=10, color=stage_colors[i], fontweight='bold')
    if i > 0:
        ax.axvline(start, color='#444', linestyle=':', linewidth=0.8)

# Annotate JH drop
ax.annotate('JH drops\n(triggers\npupation)', xy=(52, 0.4), xytext=(60, 0.7),
            fontsize=10, color='#ef4444', fontweight='bold',
            arrowprops=dict(arrowstyle='->', color='#ef4444'))

ax.set_ylabel('Relative Hormone Level', color='white', fontsize=12)
ax.set_title('Hormone Dynamics Through Metamorphosis', color='#a3e635', fontweight='bold', fontsize=14)
ax.legend(fontsize=11, facecolor='#111', edgecolor='#333', labelcolor='white', loc='upper right')
ax.set_facecolor('#111')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15)
ax.set_xlim(0, 100)
ax.set_ylim(0, 1.15)
for spine in ax.spines.values():
    spine.set_color('#333')

# ---- Panel 2: JH/Ecdysone ratio and developmental decisions ----
ax = axes[1]
ratio = jh / (ecdysone + 0.05)  # avoid division by zero
ratio_clipped = np.clip(ratio, 0, 10)

ax.plot(t, ratio_clipped, color='#e879f9', linewidth=2, label='JH / Ecdysone ratio')
ax.axhline(2.0, color='#f97316', linestyle='--', linewidth=1, alpha=0.6)
ax.axhline(0.3, color='#22d3ee', linestyle='--', linewidth=1, alpha=0.6)

ax.fill_between(t, 2.0, 10, alpha=0.08, color='#4ade80')
ax.fill_between(t, 0.3, 2.0, alpha=0.08, color='#f97316')
ax.fill_between(t, 0, 0.3, alpha=0.08, color='#22d3ee')

ax.text(20, 7, 'Larval molt zone', fontsize=11, color='#4ade80')
ax.text(48, 1.0, 'Pupal molt zone', fontsize=11, color='#f97316')
ax.text(80, 0.1, 'Adult eclosion zone', fontsize=11, color='#22d3ee')

ax.set_ylabel('JH / Ecdysone Ratio', color='white', fontsize=12)
ax.set_title('Developmental Decision: JH/Ecdysone Ratio', color='#a3e635', fontweight='bold', fontsize=14)
ax.legend(fontsize=11, facecolor='#111', edgecolor='#333', labelcolor='white')
ax.set_facecolor('#111')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15)
ax.set_xlim(0, 100)
ax.set_ylim(0, 10)
for spine in ax.spines.values():
    spine.set_color('#333')

# ---- Panel 3: Body mass and imaginal disc growth ----
ax = axes[2]

# Body mass: step-wise growth through larval instars, drops during pupation
mass = np.zeros_like(t)
for i, (start, end) in enumerate(zip(stage_bounds[:-1], stage_bounds[1:])):
    mask = (t >= start) & (t < end)
    if i < 5:  # larval instars
        t_local = (t[mask] - start) / (end - start)
        mass[mask] = (i + 1) * 0.15 + 0.15 * t_local**0.8
    elif i == 5:  # prepupa - slight decrease
        mass[mask] = 0.9 - 0.1 * (t[mask] - start) / (end - start)
    elif i == 6:  # pupa - mass loss from histolysis
        mass[mask] = 0.8 - 0.15 * (t[mask] - start) / (end - start)
    else:  # adult
        mass[mask] = 0.65

ax.plot(t, mass, color='#a3e635', linewidth=2.5, label='Body mass')

# Imaginal disc size (exponential growth during L3-L5, then rapid expansion at pupation)
disc_size = np.zeros_like(t)
disc_mask = t > 20
disc_size[disc_mask] = 0.05 * np.exp(0.05 * (t[disc_mask] - 20))
disc_size[t > 58] = disc_size[t <= 58].max()  # plateau during pupation
pupa_mask = (t > 65) & (t < 85)
disc_size[pupa_mask] = disc_size[t <= 58].max() * (1 + 2.0 * (t[pupa_mask] - 65) / 20)
disc_size[t >= 85] = disc_size[pupa_mask].max()
disc_size = disc_size / disc_size.max() * 0.9

ax.plot(t, disc_size, color='#22d3ee', linewidth=2, linestyle='--', label='Imaginal disc size')

# Histolysis/histogenesis markers
ax.axvspan(58, 72, alpha=0.1, color='#ef4444')
ax.text(63, 0.95, 'Histolysis', fontsize=10, color='#ef4444', fontweight='bold')
ax.axvspan(72, 85, alpha=0.1, color='#4ade80')
ax.text(74, 0.95, 'Histogenesis', fontsize=10, color='#4ade80', fontweight='bold')

ax.set_xlabel('Developmental Time (arbitrary units)', color='white', fontsize=12)
ax.set_ylabel('Relative Size', color='white', fontsize=12)
ax.set_title('Body Mass and Imaginal Disc Growth', color='#a3e635', fontweight='bold', fontsize=14)
ax.legend(fontsize=11, facecolor='#111', edgecolor='#333', labelcolor='white')
ax.set_facecolor('#111')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15)
ax.set_xlim(0, 100)
ax.set_ylim(0, 1.05)
for spine in ax.spines.values():
    spine.set_color('#333')

fig.patch.set_facecolor('#0a0a0a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()

print("Metamorphosis Hormonal Control Summary")
print("=" * 55)
print("\nKey Hormones:")
print("  JH (Juvenile Hormone): from corpora allata")
print("  20-HE (Ecdysone): from prothoracic glands")
print("\nDecision Matrix:")
print("  High JH + Ecdysone pulse -> Larval molt")
print("  Low JH  + Ecdysone pulse -> Pupal molt")
print("  No JH   + Ecdysone pulse -> Adult eclosion")
print("\nCritical events:")
print("  JH drop at L5 triggers irreversible pupation")
print("  Imaginal discs expand 100x during pupation")
print("  Histolysis recycles ~80% of larval tissue")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation 2: Morphogen Gradient in Imaginal Discs

Models the diffusion-degradation morphogen gradient (Dpp) in the Drosophila wing imaginal disc, showing how the decay length determines pattern precision.

Morphogen Gradient Patterning in Wing Imaginal Disc

Python

script.py185 lines

import numpy as np
import matplotlib.pyplot as plt
import matplotlib
matplotlib.use('Agg')

fig, axes = plt.subplots(2, 2, figsize=(14, 11))

# ---- Panel 1: Morphogen concentration profiles ----
ax = axes[0, 0]
x = np.linspace(-100, 100, 1000)  # distance from source (micrometers)

# Different decay lengths
lambdas = [10, 20, 30, 50]  # micrometers
colors = ['#ef4444', '#f97316', '#a3e635', '#22d3ee']

for lam, color in zip(lambdas, colors):
    C = np.exp(-np.abs(x) / lam)
    ax.plot(x, C, color=color, linewidth=2, label=f'lambda = {lam} um')

# Threshold lines for fate specification
ax.axhline(0.5, color='#fbbf24', linestyle='--', linewidth=1, alpha=0.5)
ax.axhline(0.1, color='#e879f9', linestyle='--', linewidth=1, alpha=0.5)
ax.text(75, 0.52, 'Threshold 1 (vein)', fontsize=9, color='#fbbf24')
ax.text(75, 0.12, 'Threshold 2 (margin)', fontsize=9, color='#e879f9')

ax.axvline(0, color='white', linestyle=':', linewidth=0.8, alpha=0.4)
ax.text(2, 0.95, 'Source', fontsize=10, color='white')

ax.set_xlabel('Distance from Source (um)', color='white', fontsize=11)
ax.set_ylabel('Relative Concentration C/C0', color='white', fontsize=11)
ax.set_title('Morphogen Gradient Profiles', color='#a3e635', fontweight='bold', fontsize=13)
ax.legend(fontsize=10, facecolor='#111', edgecolor='#333', labelcolor='white')
ax.set_facecolor('#111')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15)
for spine in ax.spines.values():
    spine.set_color('#333')

# ---- Panel 2: Time evolution of gradient establishment ----
ax = axes[0, 1]
x_pos = np.linspace(0, 80, 500)
D = 0.1  # um^2/s (Dpp diffusion)
k = 1e-4  # 1/s (degradation rate)
lam_ss = np.sqrt(D / k)  # steady-state decay length

times = [60, 300, 900, 3600, 10800]  # seconds
time_labels = ['1 min', '5 min', '15 min', '1 hr', '3 hr (steady state)']
time_colors = ['#ef4444', '#f97316', '#fbbf24', '#a3e635', '#22d3ee']

for t_val, label, color in zip(times, time_labels, time_colors):
    # Approximate solution to diffusion-degradation PDE
    # C(x,t) = C_ss(x) * (1 - exp(-k*t)) + transient
    # Simplified: use error function approach
    C_ss = np.exp(-x_pos / lam_ss)
    diffusion_front = np.sqrt(4 * D * t_val)
    C_transient = 0.5 * (1 + np.vectorize(lambda xi: np.math.erf((diffusion_front - xi) / max(diffusion_front * 0.3, 0.01)))(x_pos))
    C = C_ss * (1 - np.exp(-k * t_val)) * C_transient + C_ss * (1 - C_transient) * np.exp(-k * t_val) * 0.1
    C = np.clip(C, 0, 1)
    # Better approximation: interpolate between transient and steady state
    alpha_t = 1 - np.exp(-k * t_val)
    C_approx = alpha_t * C_ss + (1 - alpha_t) * np.exp(-x_pos**2 / (4 * D * t_val + 1e-10))
    C_approx = np.clip(C_approx / C_approx[0] if C_approx[0] > 0 else C_approx, 0, 1)
    ax.plot(x_pos, C_approx, color=color, linewidth=2, label=label)

ax.set_xlabel('Distance from Source (um)', color='white', fontsize=11)
ax.set_ylabel('C / C0', color='white', fontsize=11)
ax.set_title('Gradient Establishment Over Time', color='#a3e635', fontweight='bold', fontsize=13)
ax.legend(fontsize=9, facecolor='#111', edgecolor='#333', labelcolor='white')
ax.set_facecolor('#111')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15)
for spine in ax.spines.values():
    spine.set_color('#333')

# ---- Panel 3: Imaginal disc growth model ----
ax = axes[1, 0]

# Logistic growth of imaginal disc cell number
# dN/dt = r*N*(1 - N/K) modified by ecdysone-dependent growth factor
t_dev = np.linspace(0, 120, 1000)  # hours of L3 development

K = 50000  # carrying capacity (cells)
r = 0.05   # growth rate (1/hr)
N0 = 50    # initial cell number

# Without ecdysone modulation
N_simple = K / (1 + (K/N0 - 1) * np.exp(-r * t_dev))

# With ecdysone growth pulses
# Ecdysone enhances proliferation rate
ecdysone_mod = 1.0 + 0.5 * np.exp(-((t_dev - 40)/8)**2) + 0.8 * np.exp(-((t_dev - 90)/8)**2)
N_mod = np.zeros_like(t_dev)
N_mod[0] = N0
dt = t_dev[1] - t_dev[0]
for i in range(1, len(t_dev)):
    dN = r * ecdysone_mod[i] * N_mod[i-1] * (1 - N_mod[i-1]/K)
    N_mod[i] = N_mod[i-1] + dN * dt

ax.semilogy(t_dev, N_simple, color='#888', linewidth=1.5, linestyle='--', label='Logistic (no hormones)')
ax.semilogy(t_dev, N_mod, color='#a3e635', linewidth=2.5, label='With ecdysone modulation')
ax.axhline(K, color='#f97316', linestyle=':', linewidth=1, alpha=0.5)
ax.text(5, K*1.2, f'K = {K:,} cells', fontsize=10, color='#f97316')

# Mark ecdysone pulses
ax.axvspan(32, 48, alpha=0.08, color='#fbbf24')
ax.axvspan(82, 98, alpha=0.08, color='#fbbf24')
ax.text(40, 100, 'Ecdysone\npulse', fontsize=9, color='#fbbf24', ha='center')
ax.text(90, 100, 'Ecdysone\npulse', fontsize=9, color='#fbbf24', ha='center')

ax.set_xlabel('Developmental Time (hours in L3)', color='white', fontsize=11)
ax.set_ylabel('Cell Number', color='white', fontsize=11)
ax.set_title('Imaginal Disc Growth', color='#a3e635', fontweight='bold', fontsize=13)
ax.legend(fontsize=10, facecolor='#111', edgecolor='#333', labelcolor='white')
ax.set_facecolor('#111')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15)
for spine in ax.spines.values():
    spine.set_color('#333')

# ---- Panel 4: Caspase activation bistable switch ----
ax = axes[1, 1]

# Nullcline analysis of caspase activation
# dx/dt = k1*S + k2*x^2/(Km^2 + x^2) - kd*x
# where x = [DrICE*], S = [Dronc*] (treated as parameter)

x_casp = np.linspace(0, 1, 500)
k1 = 0.1
k2_vals = [0.3, 0.5, 0.8]
Km = 0.3
kd = 0.5

S_values = [0.0, 0.2, 0.5, 0.8]
S_colors = ['#888', '#f97316', '#a3e635', '#22d3ee']

for S, color in zip(S_values, S_colors):
    # Production rate
    production = k1 * S + 0.5 * x_casp**2 / (Km**2 + x_casp**2)
    # Degradation rate
    degradation = kd * x_casp
    # Net rate
    net = production - degradation

ax.plot(x_casp, net, color=color, linewidth=2, label=f'Dronc* = {S:.1f}')
    # Mark zeros (steady states)
    sign_changes = np.where(np.diff(np.sign(net)))[0]
    for idx in sign_changes:
        stability = 'o' if net[idx+1] - net[idx] < 0 else 'x'
        marker_color = '#4ade80' if stability == 'o' else '#ef4444'
        ax.plot(x_casp[idx], 0, marker=stability, color=marker_color, markersize=8, markeredgewidth=2)

ax.axhline(0, color='white', linewidth=0.5, alpha=0.3)
ax.set_xlabel('[DrICE*] (normalized)', color='white', fontsize=11)
ax.set_ylabel('d[DrICE*]/dt', color='white', fontsize=11)
ax.set_title('Caspase Bistable Switch', color='#a3e635', fontweight='bold', fontsize=13)
ax.legend(fontsize=10, facecolor='#111', edgecolor='#333', labelcolor='white')
ax.set_facecolor('#111')
ax.tick_params(colors='white')
ax.grid(True, alpha=0.15)
ax.text(0.7, 0.15, 'ALIVE', fontsize=12, color='#4ade80', fontweight='bold')
ax.text(0.05, -0.08, 'Apoptotic', fontsize=12, color='#ef4444', fontweight='bold')
for spine in ax.spines.values():
    spine.set_color('#333')

fig.patch.set_facecolor('#0a0a0a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()

print("Morphogen Gradient and Disc Growth Analysis")
print("=" * 55)
print(f"\nDpp diffusion coefficient: D = 0.1 um^2/s")
print(f"Degradation rate: k = 1e-4 /s")
print(f"Decay length: lambda = sqrt(D/k) = {np.sqrt(0.1/1e-4):.1f} um")
print(f"  (approximately {np.sqrt(0.1/1e-4)/4:.0f} cell diameters)")
print(f"\nImaginal disc growth (Drosophila wing disc):")
print(f"  Initial cells: ~{N0}")
print(f"  Final cells: ~{K:,}")
print(f"  Growth factor: {K/N0:.0f}x")
print(f"  Duration: ~120 hours (L3 instar)")
print(f"\nCaspase switch properties:")
print(f"  Positive feedback creates bistability")
print(f"  Once Dronc* exceeds threshold, DrICE activation is irreversible")
print(f"  Ensures all-or-nothing apoptotic commitment")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

Riddiford, L.M. (1993). Hormones and Drosophila development. In The Development of Drosophila melanogaster (M. Bate and A. Martinez Arias, eds.), pp. 899-939. Cold Spring Harbor Laboratory Press.
Truman, J.W. and Riddiford, L.M. (1999). The origins of insect metamorphosis. Nature, 401(6752), 447-452.
Nijhout, H.F. (1994). Insect Hormones. Princeton University Press.
Koyama, T., Texada, M.J., Halberg, K.A. and Rewitz, K. (2020). Metabolism and growth adaptation to environmental conditions in Drosophila. Cellular and Molecular Life Sciences, 77(22), 4523-4551.
Wartlick, O., Kicheva, A. and Gonzalez-Gaitan, M. (2009). Morphogen gradient formation. Cold Spring Harbor Perspectives in Biology, 1(3), a001255.
Kicheva, A., Pantazis, P., Bollenbach, T., Kalaidzidis, Y., Bittig, T., Julicher, F. and Gonzalez-Gaitan, M. (2007). Kinetics of morphogen gradient formation. Science, 315(5811), 521-525.
Fuchs, Y. and Steller, H. (2011). Programmed cell death in animal development and disease. Cell, 147(4), 742-758.
Rolff, J., Johnston, P.R. and Reynolds, S. (2019). Complete metamorphosis of insects. Philosophical Transactions of the Royal Society B, 374(1783), 20190063.
Beira, J.V. and Paro, R. (2016). The legacy of Drosophila imaginal discs. Chromosoma, 125(4), 573-592.
Yang, A.S. (2001). Modularity, evolvability, and adaptive radiations: a comparison of the hemi- and holometabolous insects. Evolution and Development, 3(2), 59-72.

<- Module 1: Flight Mechanics Module 3: Sensory Systems ->